On Machines in General. 125 



IF attraction was always constant like ordinary gravity, 

 but, if directed towards a fixed point, placed at a finite 

 distance, we might easily conclude from the preceding prin- 

 ciple, that in the case of equilibrium, the sum of the mo- 

 menta of the bodies of the system, relatively to this fixed 

 point, is a maximum, i. e. the sum of the products of each 

 mass, by its distance to the fixed point, is less when there 

 is an equilibrium, than if the system was placed in any 

 other given situation. 



If the attraction towards the fixed point, instead of being 

 constant, was proportional to the distances from this body 

 to this fixed point, we might conclude in the same way 

 that the sum of the products of each mass by the 9quare of 

 the distance to this fixed .point, is a maximum. 



We know that the sum of the products of each mass, by 

 the square of its distance to any fixed point, is equal to the 

 sum of the products of each mass, by the square of its di- 

 stance to the centre of gravity; plus, to the product of the 

 total mass, by the square of the distance from the cen- 

 tre of gravity to this fixed point : (this is a well-known pro- 

 position in geometry, and easily proved ;) thus, in the case 

 of attraction under examination, the sum of these two quan- 

 tities should, in the case of equilibrium, be a maximum, i. e. 

 its differential is equal to zero. Let us supposes f° r instance, 

 that all the parts of the system are connected with each 

 other, so as to form only one body, and that this body is 

 suspended by its centre of gravity, so that this point is fixed; 

 it is clear that each of the quantities mentioned will be con- 

 stant ; i. e. will remain the same, whatever situation we 

 give to this body, and the differential of their sum will con- 

 sequently be null : thus there will be equilibrium ; i. e. if 

 all the particles of a body are attracted towards a fixed point, 

 proportional to their distances to this point, and if we sus- 

 pend this body by its centre of gravity, it will remain in 

 equilibrium precisely as in the case of ordinary gravity. It 

 must not be concluded from this, however, that in a machine 

 to which several bodies are applied, attracted towards a fixed 

 point, in ratio of the distances, the position of equilibrium 

 was that at which the centre of gravity of the system would 



be 



